You are given 3 boxes - one labeled "Apples", one labeled "Bananas", and one labeled "Apples and Bananas". You are told that the labels on the boxes have been completely mismatched, i.e. none of the three labels is put on its correct box. How can you open just one box and pick a random fruit from it, so that after seeing the fruit, you can guess correctly the contents of every box out of the three?

Open the box labeled "Apples and Bananas". If you pick a banana from it, then the box labeled "Bananas" will contain apples, and therefore the box labeled "Apples" will contain apples and bananas. Similarly, if you pick an apple from it, then the box labeled "Apples" will contain bananas, and therefore the box labeled "Bananas" will contain apples and bananas".